Here is a collection of puzzles about Sam's shop sent in by club
members. Perhaps you can make up more puzzles, find formulas or
find general methods.

Handshakes

Stage: 3 Challenge Level:

Seven mathematicians met up one week.

The first mathematician shook hands with all the others.
The second one shook hands with all the others apart from the first
one (since they had already shaken hands).
The third one shook hands with all the others apart from the first
and the second mathematicians, and so on, until everyone had shaken
hands with everyone else.

How many handshakes were there altogether?

The next week, eight mathematicians
met. How many handshakes took place this time?

The following week, there were nine mathematicians...

Sam is trying to work out how many handshakes there would be
if 20 mathematicians met. He says that since each mathematician
shakes hands 19 times, there must be $20 \times 19$ handshakes
altogether. Helen disagrees; she worked out $19 + 18 + 17 + ... + 2
+ 1$ and got a different answer. What is wrong with Sam's
reasoning? How should he modify his method?

One day, 161 mathematicians met. How many handshakes took
place this time?

Can you describe a quick way of working out the number of
handshakes for any size of mathematical gathering?

Could there be exactly 4851 handshakes at a gathering where
everyone shakes hands? How many mathematicians would there
be?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.